Double Cosets Record Two-Sided Symmetry

A double coset H g K identifies elements after independent multiplication from a subgroup on each side. The formulas are more useful when each symbol has a job rather than merely decorating the theorem.

The data

A permutation in $S_n$ is best read through its disjoint cycle type. Group actions then translate algebra into orbits $Gx$, stabilisers $G_x$, and fixed-point counts.

$$ HgK=\{hgk:h\in H,\ k\in K\} $$

I read the first line as input and the second as output. The symbols $\forall$ and $\exists$ are not interchangeable, and neither may be upgraded silently to $\Longleftrightarrow$.

$$ |HgK|=\frac{|H|\,|K|}{|H\cap gKg^{-1}|} $$

Derivation

A worked instance is useful here because it exposes every index that the compressed statement hides.

$$ H\backslash G/K=\bigsqcup_{[g]}\{HgK\},\qquad \mathbf1_H*\mathbf1_K(g)=|H\cap gK^{-1}| $$

Invariant content

The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.

$$ \begin{aligned} \mathsf{D}\;&:\quad HgK=\{hgk:h\in H,\ k\in K\},\\[5pt] \mathsf{C}\;&:\quad |HgK|=\frac{|H|\,|K|}{|H\cap gKg^{-1}|}. \end{aligned} $$

Scope

Cycle notation suppresses fixed points, so the ambient symmetric group still matters. The cycle $(1\,2\,3)$ in $S_3$ and in $S_8$ has different centralisers.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] |HgK|=\frac{|H|\,|K|}{|H\cap gKg^{-1}|} \end{gathered}} $$

I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.

This article was posted on Wed 30 December 2020. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.

Double Cosets Record Two-Sided Symmetry by Vicfred

The data

Derivation

Invariant content

Scope